Final answer:
To evaluate the derivative of the dot product (r(t) · r1(t)), apply the Product Rule for vector functions, which involves calculating the derivatives of the individual functions and then evaluating the sum of both dot products at the specified value of t.
Step-by-step explanation:
The question pertains to taking the derivative of the dot product between two vector functions r(t) and r1(t). To find the derivative, d/dt of (r(t) · r1(t)), we apply the Product Rule, which in the context of vector functions, states that the derivative of the dot product of two vector functions is equal to the derivative of the first function dotted with the second function plus the first function dotted with the derivative of the second function.
Given that
r(2) = (2,1,0), and r'(2) = (1,4,3), we would compute the derivative at t=2 using the fact that r1(t) = (t^2, t^3, 8t). Therefore, r1'(t) at t=2 would need to be computed using the individual derivatives of the components of r1(t), which are 2t, 3t^2, and 8 respectively. Substituting t=2 in r1(t) and r1'(t), we would then calculate r'(2) · r1(2) + r(2) · r1'(2) to evaluate the dot product derivative at t = 2.